Let X1,n[less-than-or-equals, slant]...[less-than-or-equals, slant]Xn,n be the order statistics of n independent random variables with a common distribution function F and let kn be positive numbers such that kn --> [infinity] and . With suitable centering and norming, we investigate the weak convergence of the intermediate-sum process [summation operator]i=[left ceiling]akn[right ceiling]+1[left ceiling]tkn[right ceiling]Xn+1-i,n, a [less-than-or-equals, slant] t [less-than-or-equals, slant] b, where 0 < a < b < [infinity], and the weak convergence of the extreme-sum process [summation operator]i=1[left ceiling]tkn[right ceiling]Xn+1-i,n, 0 [less-than-or-equals, slant] t [less-than-or-equals, slant] b. Convergence is with respect to the supremum norm and can take place along a subsequence of the positive integers {n}.
